Quadrilaterals and other

Properties of Geometric Figures, SM-Bank 039

Determine the value of \(x^{\circ}\) in the quadrilateral above, giving reasons for your answer. (2 marks)

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\(30^{\circ}\)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\)	\(=2x + 3x + 4x + 2x \)
\(12x^{\circ}\)	\(=360\)
\(x^{\circ}\)	\(=\dfrac{360}{12}\)
	\(=30^{\circ}\)

Properties of Geometric Figures, SM-Bank 041

A pentagon is pictured below.

By drawing triangles from one vertex, or otherwise, calculate the sum of the internal angles of a pentagon. (1 mark)

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Determine the value of \(x^{\circ}\) in the pentagon. (2 marks)

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i. \(540^{\circ}\)

ii. \(110^{\circ}\)

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i. \(\text{Pentagon can be divided into 3 triangles (from one chosen vertex).}\)

\(\text{Sum of internal angles}\ = 3 \times 180 = 540^{\circ}\)

ii. \(540\)	\(=x + 2 \times 90 + 135+115 \)
\(540\)	\(=x+430\)
\(x^{\circ}\)	\(=540-430\)
	\(=110^{\circ}\)

Properties of Geometric Figures, SM-Bank 040

Determine the value of the two unknown angles in the quadrilateral above, giving reasons for your answer. (3 marks)

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\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\)	\(=5x+3x+79+105 \)
\(8x\)	\(=360-184\)
\(x^{\circ}\)	\(=\dfrac{176}{8}\)
	\(=22^{\circ}\)

\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

Show Worked Solution

\(\text{Angle sum of quadrilaterals = 360°:} \)

\(360\)	\(=5x+3x+79+105 \)
\(8x\)	\(=360-184\)
\(x^{\circ}\)	\(=\dfrac{176}{8}\)
	\(=22^{\circ}\)

\(\text{Unknown angle 1}\ = 3 \times 22 = 66^{\circ}\)

\(\text{Unknown angle 2}\ = 5 \times 22 = 110^{\circ}\)

Properties of Geometric Figures, SM-Bank 038

\(ABCD\) is a trapezium.

Determine the value of \(x^{\circ}\), giving reasons for your answer. (2 marks)

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\(67^{\circ}\)

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\(DA \parallel CB \ \ (ABCD\ \text{is a trapezium}) \)

\(x+113\)	\(=180\ \ \text{(cointerior angles)} \)
\(x^{\circ}\)	\(=180-113\)
	\(=67^{\circ}\)

Properties of Geometric Figures, SM-Bank 037

\(ABCD\) is a trapezium.

Determine the value of \(x^{\circ}\), giving reasons for your answer. (2 marks)

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\(83^{\circ}\)

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\(AD \parallel BC \ \ (ABCD\ \text{is a trapezium}) \)

\(x+97\)	\(=180\ \ \text{(cointerior angles)} \)
\(x^{\circ}\)	\(=180-97\)
	\(=83^{\circ}\)

Properties of Geometric Figures, SM-Bank 036

\(ABCD\) is a parallelogram.

Determine the value of \(a^{\circ}\), giving reasons for your answer. (2 marks)

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\(AB \parallel DC \ \ \text{(opposite sides of parallelogram)} \)

\(a+125\)	\(=180\ \ \text{(cointerior angles)} \)
\(a^{\circ}\)	\(=180-125\)
	\(=55^{\circ}\)

Show Worked Solution

\(AB \parallel DC \ \ \text{(opposite sides of parallelogram)} \)

\(a+125\)	\(=180\ \ \text{(cointerior angles)} \)
\(a^{\circ}\)	\(=180-125\)
	\(=55^{\circ}\)

Properties of Geometrical Figures, SM-Bank 035

Find the value of \(a^{\circ}\) in the diagram below. (2 marks)

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\(132^{\circ}\)

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\(\text{Since there are 360° in a quadrilateral:}\)

\(360\)	\(=a+62+85+81\)
\(360\)	\(=a+228\)
\(a^{\circ}\)	\(=360-228\)
	\(=132^{\circ}\)

Properties of Geometrical Figures, SM-Bank 034

Find the value of \(x^{\circ}\) in the diagram below. (2 marks)

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\(81^{\circ}\)

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\(\text{Since there are 360° in a quadrilateral:}\)

\(360\)	\(=x+98+108+73\)
\(360\)	\(=x+279\)
\(x^{\circ}\)	\(=360-279\)
	\(=81^{\circ}\)

Properties of Geometric Figures, SM-Bank 033

Find the value of \(a^{\circ}\) in the diagram below. (2 marks)

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\(60^{\circ}\)

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\(\angle BCD\ \text{(reflex)} = 360-130=230^{\circ}\)

\(\text{Since there are 360° in a quadrilateral:}\)

\(360\)	\(=a+40+230+30\)
\(360\)	\(=a+300\)
\(a^{\circ}\)	\(=360-300\)
	\(=60^{\circ}\)

Properties of Geometric Figures, SM-Bank 032

Find the value of \(x^{\circ}\) in the diagram below. (2 marks)

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\(110^{\circ}\)

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\(\text{Since there are 360° in a quadrilateral:}\)

\(360\)	\(=x+55+105+90\)
\(360\)	\(=x+250\)
\(x^{\circ}\)	\(=360-250\)
	\(=110^{\circ}\)

Properties of Geometric Figures, SM-Bank 031 MC

The diagram of a quadrilateral is shown below.

Which name below does not refer to the quadrilateral in the diagram?

quadrilateral \(CDAB\)
quadrilateral \(BCDA\)
quadrilateral \(CBAD\)
quadrilateral \(CBDA\)

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\(D\)

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\(\text{Vertices need to be named in order (either clockwise or counter clockwise)}\)

\(CBDA\ \text{is not correct as vertex}\ B\ \text{and}\ D\ \text{are not adjacent.}\)

\(\Rightarrow D\)

Properties of Geometrical Figures, SM-Bank 030

\(ABCDE\) is a pentagon.

Using point \(A\) as one vertex, divide the pentagon into the maximum number of triangles possible. (1 mark)

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Using the angle sum of one triangle, show that the sum of the internal angles of the pentagon is 540°. (2 marks)

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ii. \(ABCDE\ \text{can be divided into 3 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCDE = 3 \times 180^{\circ} = 540^{\circ}\)

Show Worked Solution

ii. \(ABCDE\ \text{can be divided into 3 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCDE = 3 \times 180^{\circ} = 540^{\circ}\)

Properties of Geometrical Figures, SM-Bank 029

Divide quadrilateral \(ABCD\) into triangles and using the angle sum of one triangle, determine the sum of the internal angles of a quadrilateral. (2 marks)

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\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Show Worked Solution

\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Properties of Geometric Figures, SM-Bank 028

Divide quadrilateral \(ABCD\) into triangles and using the angle sum of one triangle, determine the sum of the internal angles of a quadrilateral. (2 marks)

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\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Show Worked Solution

\(ABCD\ \text{can be divided into 2 triangles.}\)

\(\text{Angle sum of a triangle = 180°}\)

\(\text{Angle sum of}\ ABCD = 2 \times 180^{\circ} = 360^{\circ}\)

Properties of Geometric Figures, SM-Bank 044

Complete the table below by placing a tick or a cross in the appropriate box to indicate which properties belong to different quadrilaterals. (3 marks)

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Rhombus} & \textbf{Trapezium} & \textbf{Rectangle} \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are equal} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}

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\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Rhombus} & \textbf{Trapezium} & \textbf{Rectangle} \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \cross \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are equal} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Rhombus} & \textbf{Trapezium} & \textbf{Rectangle} \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \cross \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are equal} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\end{array}

Properties of Geometric Figures, SM-Bank 043

Complete the table below by placing a tick or a cross in the appropriate box to indicate which properties belong to different quadrilaterals. (3 marks)

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Square} & \textbf{Kite} & \textbf{Parallelogram} \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals bisect each other} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Two pairs of equal adjacent sides} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}

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\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Square} & \textbf{Kite} & \textbf{Parallelogram} \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals bisect each other} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Two pairs of equal adjacent sides} \rule[-1ex]{0pt}{0pt} & \checkmark & \checkmark & \cross \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Square} & \textbf{Kite} & \textbf{Parallelogram} \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals bisect each other} \rule[-1ex]{0pt}{0pt} & \checkmark & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Two pairs of equal adjacent sides} \rule[-1ex]{0pt}{0pt} & \checkmark & \checkmark & \cross \\
\hline
\end{array}

Properties of Geometrical Figures, SM-Bank 042

Complete the table below by placing a tick or a cross in the appropriate box to indicate which properties belong to different quadrilaterals. (3 marks)

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Trapezium} & \textbf{Rectangle} & \textbf{Rhombus} \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}

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\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Trapezium} & \textbf{Rectangle} & \textbf{Rhombus} \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \cross \\
\hline
\end{array}

Show Worked Solution

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ \ \ \ \ \textbf{Property} \rule[-1ex]{0pt}{0pt} & \textbf{Trapezium} & \textbf{Rectangle} & \textbf{Rhombus} \\
\hline
\rule{0pt}{2.5ex} \text{Opposite sides are parallel} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Diagonals are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \cross & \checkmark \\
\hline
\rule{0pt}{2.5ex} \text{Adjacent sides are perpendicular} \rule[-1ex]{0pt}{0pt} & \cross & \checkmark & \cross \\
\hline
\end{array}

Properties of Geometrical Figures, SM-Bank 010

A six sided figure is drawn below.

What is the sum of the six interior angles? (2 marks)

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`720^@`

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`text(Reflex angle) = 360-90 = 270^@`

`:.\ text(Sum of interior angles)`

`= (270 xx 2) + (30 xx 2) + (60 xx 2)`

`= 720^@`

Properties of Geometric Figures, SM-Bank 009 MC

Choon drew a shape. It had one pair of parallel sides and two internal reflex angles.

Which of these could be Choon's shape?


A	B	C	D

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\(D\)

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\(\text{Internal reflex angles are greater than 180°.}\)

\(\Rightarrow D\)

Properties of Geometric Figures, SM-Bank 007

In the diagram \(AB\) is a straight line.

Calculate the size of the angle marked \(x^{\circ}\), giving reasons for your answer. (3 marks)

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\(\text{Equilateral triangle}\ \ \Rightarrow\ \ \text{all angles}\ = 60^{\circ}\)

\(\text{Vertically opposite angles of 60° are equal (see diagram)}\)

\(x^{\circ} = 180-(90+60) = 30^{\circ}\ \ \text{(180° in straight line)}\)

Show Worked Solution

\(\text{Equilateral triangle}\ \ \Rightarrow\ \ \text{all angles}\ = 60^{\circ}\)

\(\text{Vertically opposite angles of 60° are equal (see diagram)}\)

\(x^{\circ} = 180-(90+60) = 30^{\circ}\ \ \text{(180° in straight line)}\)

Properties of Geometric Figures, SM-Bank 006

Pablo creates a design that is made up of 3 rectangles and 2 straight lines, as shown below.

What is the size of angle \(x^{\circ}\)? (3 marks)

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\(\text{135 degrees}\)

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\(\text{Isosceles triangle}\ \ \Rightarrow\ \ \text{angles opposite equal sides are equal} \)

\(\text{Since there is 180° in a straight line:}\)

\(x + 45\)	\(= 180\)
\(x^{\circ}\)	\(= 135^{\circ}\)

Properties of Geometrical Figures, SM-Bank 005 MC

A regular hexagon is folded in half along the dotted line.

The folded shape can also be called a

pentagon
hexagon
quadrilateral
nonagon

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`C`

Show Worked Solution

`text{The folded shape is a quadrilateral (four sides).}`

`=>C`

Special Properties, SMB-013 MC

Which statement is always true?

Scalene triangles have two angles that are equal.
All angles in a parallelogram are equal.
The opposite sides of a trapezium are equal in length.
The diagonals of a rhombus are perpendicular to each other.

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`D`

Show Worked Solution

`text{Consider each option:}`

`A:\ \text{Isosceles (not scalene) have two equal angles.}`

`B:\ \text{Only opposite angles in a parallelogram are equal.}`

`C:\ \text{At least one pair of opposite sides of a trapezium are not equal.}`

`D:\ \text{Rhombuses have perpendicular diagonals.}`

`=>D`

Special Properties, SMB-011 MC

The diagonals of which shape below cross at right-angles?

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\(A\)

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`text(A rhombus has diagonals that cross at right-angles.)`

\(\Rightarrow A \)

Special Properties, SMB-010 MC

Which of these are always equal in length?

the diagonals of a rhombus
the diagonals of a parallelogram
the opposite sides of a parallelogram
the opposite sides of a trapezium

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`C`

Show Worked Solution

`text{Consider each option:}`

`A:\ \text{rhombus diagonals are perpendicular but not always equal}`

`B:\ \text{parallelogram diagonals not always equal (see below)}`

`C:\ \text{always true (see above)}`

`D:\ \text{at least 1 pair of opposite sides of a trapezium are not equal}`
\(\Rightarrow C\)

Special Properties, SMB-009 MC

`PQRS` is a parallelogram.

Which of these must be a property of `PQRS`?

Line `PS` is perpendicular to line `PQ`.
Line `PQ` is parallel to line `PS`.
Diagonals `PR` and `SQ` are perpendicular.
Line `PS` is parallel to line `QR`.

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`D`

Show Worked Solution

`text{By elimination:}`

`A\ \text{and}\ B\ \text{clearly incorrect.}`

`C\ \text{true if all sides are equal (rhombus) but not true for all parallelograms.}`

`text(Line)\ PS\ text(must be parallel to line)\ QR.`

`=>D`

Special Properties, SMB-007 MC

A closed shape has two pairs of equal adjacent sides.

What is the shape?

rectangle
trapezium
kite
triangle

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`C`

Show Worked Solution

`text(Kite.)`

`text{(Note that a rectangle has a pair of equal opposite sides)}`

`=>C`